Star-coloring planar graphs with high girth Daniel W. Cranston - PowerPoint PPT Presentation

Star-coloring planar graphs with high girth Daniel W. Cranston DIMACS, Rutgers dcransto@dimacs.rutgers.edu Joint with Craig Timmons and Andre Kundgen

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Gr¨ unbaum 1970] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 9.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5. Def. A star coloring is a proper vertex coloring such that the union of any two color classes induces a star forest (contains no P 4 ).

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5. Def. A star coloring is a proper vertex coloring such that the union of any two color classes induces a star forest (contains no P 4 ). [Fetin-Raspaud-Reed 2001] Thm. Every planar G has star chromatic number χ s ( G ), at most 80.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5. Def. A star coloring is a proper vertex coloring such that the union of any two color classes induces a star forest (contains no P 4 ). [Albertson-Chappell-Kierstead-K¨ undgen-Ramamurthi ’04] Thm. Every planar G has star chromatic number χ s ( G ), at most 20.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5. Def. A star coloring is a proper vertex coloring such that the union of any two color classes induces a star forest (contains no P 4 ). [Albertson-Chappell-Kierstead-K¨ undgen-Ramamurthi ’04] Thm. Every planar G has star chromatic number χ s ( G ), at most 20.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5. Def. A star coloring is a proper vertex coloring such that the union of any two color classes induces a star forest (contains no P 4 ). [Albertson-Chappell-Kierstead-K¨ undgen-Ramamurthi ’04] Thm. Every planar G has star chromatic number χ s ( G ), at most 20.

Definitions and Examples Def. An acyclic coloring is a proper vertex coloring such that the union of any two color classes induces a forest. [Borodin 1979] Thm. Every planar G has acyclic chromatic number, χ a ( G ), at most 5. Def. A star coloring is a proper vertex coloring such that the union of any two color classes induces a star forest (contains no P 4 ). [Albertson-Chappell-Kierstead-K¨ undgen-Ramamurthi ’04] Thm. Every planar G has star chromatic number χ s ( G ), at most 20.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G .

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. r Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. r Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. r Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. r Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. r Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Structural Decomposition Thm. [A-C-K-K-R] For every surface S there is a constant γ such that every graph G with girth ≥ γ embedded in S has χ s ( G ) ≤ 4. Thm. [Timmons ’07] If G is planar and has girth ≥ 14, then we can partition V ( G ) into sets I and F s.t. G [ F ] is a forest and I is a 2-independent set in G . Def. A set I is 2-independent in G if ∀ u , v ∈ I dist( u , v ) > 2. Lem. If we can partition G as in Theorem, then χ s ( G ) ≤ 4. r Pf. Choose a root in each tree of F . If v ∈ F is distance k from its root, then v gets color k (mod 3). If v ∈ I , then v gets color 3.

Reducibility Pf. Assume that G is a minimal counterexample. G must not contain any of the following subgraphs:

Reducibility Pf. Assume that G is a minimal counterexample. G must not contain any of the following subgraphs: v

Reducibility Pf. Assume that G is a minimal counterexample. G must not contain any of the following subgraphs: v Partition G − v .